Divide using synthetic division $$ ( 12x^{2} ) \div ( 3x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-1&12&0&0\\& & -12& \color{black}{12} \\ \hline &\color{blue}{12}&\color{blue}{-12}&\color{orangered}{12} \end{array} $$The solution is:
$$ \frac{ 12x^{2} }{ x+1 } = \color{blue}{12x-12} ~+~ \frac{ \color{red}{ 12 } }{ x+1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-1}&12&0&0\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-1&\color{orangered}{ 12 }&0&0\\& & & \\ \hline &\color{orangered}{12}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 12 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrr}\color{blue}{-1}&12&0&0\\& & \color{blue}{-12} & \\ \hline &\color{blue}{12}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrr}-1&12&\color{orangered}{ 0 }&0\\& & \color{orangered}{-12} & \\ \hline &12&\color{orangered}{-12}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrr}\color{blue}{-1}&12&0&0\\& & -12& \color{blue}{12} \\ \hline &12&\color{blue}{-12}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 12 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrr}-1&12&0&\color{orangered}{ 0 }\\& & -12& \color{orangered}{12} \\ \hline &\color{blue}{12}&\color{blue}{-12}&\color{orangered}{12} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 12x-12 } $ with a remainder of $ \color{red}{ 12 } $.